Equilibrium swelling

Vulcanization and cross-linking co-polymerization are the technically most relevant processes to synthesize polymer networks and gels, whereby the latter was addressed only in few computer simulation studies so far not yet covering the equilibrium degree of swelling of these networks.

This gap was closed with our work [1] in which we studied both the formation of the network structure by simulating the corresponding diffusion reaction problem and the equilibrium swelling of the final network. Quite interestingly, we observed an auto-acceleration of reaction kinetics similar to the “Trommsdorff-Norrish effect” (or “gel effect”) independent of the degree of cross-linking. This challenges the current discussion of this effect as an explicit cross-link dependence is in the core of most works as one sees straight from calling it “gel effect”.
We found that the scaling of the equilibrium degree of swelling as a function of the elastic strand length as estimated from cross-link density (as done in most experiments) is in the range of the Flory-Rehner model for equilibrium swelling [1]. However, this is just due to a fortuitous cancellation of the corrections due to inactive material and entanglements. Consideration of only one correction leads to largely unsatisfying results, however, after taking into account both corrections in an effective degree of polymerization of the elastic strands, Neff, the equilibrium swelling data of the networks follows again the classical prediction in a good approximation. In the limit of rather low cross-link density (large Neff), an additional non-affine contributions is observed as shown by the small disagreement between simulation data (points) and the Flory Rehner model (line) in the Figure. This deviation is qualitatively similar to the recently observed desinterspersion process of overlapping strands in Olympic gels [2] that might become significant in this very limit.

Olympic gels are networks that are held together solely by the mutual concatenation of cyclic polymers. It has not been possible yet to fabricate such gels in a controlled manner such that computer simulations data are the only source to test predictions for these model systems so far. With computer simulations, one can generate ideal Olympic gels made of perfectly monodisperse cyclic polymers that freely interpenetrate each other [2]. After equilibration, entanglements were switched on, which fixed all concatenations. In a subsequent step, the Olympic gels were placed into a larger compartment and swollen to equilibrium.
The equilibrium swelling data appeared quite puzzling at first sight as the classical scaling as a function of the degree of polymerization of the strands was qualitatively reversed: gels made of longer strands swell less [2]. However, a thorough investigation of the changes upon swelling revealed that there are two mechanisms at work during swelling: a desinterspersion process of non-concatenated but overlapping rings and a swelling of the concatenated rings. Consideration of both effects lead to a collapse of all simulation data and a scaling model for the equilibrium degree of swelling of Olympic gels as a function of desinterspersion.


  1. Lang, M. ; John, A. ; Sommer, J.-U.,
    Model simulations on network formation and swelling as obtained from cross-linking co-polymerization reactions
    Polymer 82 (2016) 138-155.
  2. Lang, M. ; Fischer, J. ; Werner, M. ; Sommer, J.-U.,
    Swelling of Olympic gels
    Physical Review Letters 112 (2014) 238001(5).